My travel kits are
1/ Sennheiser ME400 shotgun mic and a Tentacle Track-E, with the mic and recorder connected to a Smallrig pistol grip with cold shoe
2/ Dodotronics Highsound Compact, again with the Tentacle Track-E recorder and the Smallrig pistol grip.
You could easily switch the ME400 for the Rode VideoMic Pro, which has slightly better sensitivity, but only comes with foam windscreen, rather than a dead cat fur wind shield. Also the suspension on the Rode Mic seems a bit more delicate (although my wife hasn't damaged her mic yet).
Both mics are designed for On-Camera, so have small size format, but need to fit in a cold shoe - but this works OK with an appropriate pistol grip.
Pros for option 1: Small size (easily fits in a small camera bag), built-in suspension system (to reduce handling noise), good dead-cat cover supplied with mic (to alleviate wind noise), tough recorder (even though it is plastic, it has taken a bit of a bashing over the years - including getting wet), 32-bit float recording (so no real need to monitor and adjust recording levels), sound pretty good (in my opinion).
Cons: On-off and record a bit fiddly to operate (impossible with gloves), a bit expensive
Pros for option 2: Quite small (I carry the kit using a Black Rapid Sling, with the small dish resting behind my hip), better gain from the dish (so more suited to birds a bit further away), again 32-bit float and again reasonable sound quality.
Cons: Stereo parabola to mono recorder, so you need a stereo-to-mono jack (and loose the stereo effect), occasional problems with connections (due to stereo to mono adapter - only cheap adapters seem available), more bulky and mic feels less robust, sometimes the dish has come loose from the cold shoe (probably due to the way I carry it), a bit expensive, supply of the dish outside the EU is a bit problematic (and expect import duty on top of the website sales price).
I am looking to change the recorder I use with the parabola to the Tascam FR-AV2, which has a stereo mini jack input (unlike the otherwise similar Zoom F3). I see the problem is that this recorder can only be 'attached' to the pistol grip with a camera monitor bracket, screwed into the tripod thread - the device is 'all screen', so I cannot tape it in place like the Tentacle Track E (that has no screen!) - this could easily work loose - and also screens can be a hindrance (especially as LCDs can be vulnerable to rain and high humidity).
The combo one was covered by a post I made some time ago - see
ME400 and Tentacle Track E.